How many satellites make up the minimum operational GPS constellation?

How to get close to this number

Work from the geometry. Fixing a position in three dimensions plus time requires four simultaneous signals, and the constellation must guarantee that from anywhere on Earth at any moment. Covering a sphere with that guarantee takes several orbital planes with several satellites each, which multiplies out to a couple of dozen.

Why four signals rather than three is the part worth understanding. Three would fix a position in space if you had a perfect clock, but a receiver does not — so the fourth satellite solves for the receiver's clock error. Every phone contains a device that continuously calculates its own time offset against atomic clocks in orbit, which is a genuinely extraordinary thing to be doing in a pocket.

The operational figure is higher than the design minimum, as the explanation notes: usually more than thirty active. Spares improve accuracy and cover failures, and the baseline is the number below which the guarantee breaks rather than the number actually flying. Design minimums and operational counts are different specifications, and infrastructure questions almost always mean one or the other specifically.